

A332742


Number of nonunimodal negated permutations of a multiset whose multiplicities are the prime indices of n.


12



0, 0, 0, 0, 0, 1, 0, 2, 3, 2, 0, 8, 0, 3, 7, 16, 0, 24, 0, 16, 12, 4, 0, 52, 16, 5, 81, 26, 0, 54, 0, 104, 18, 6, 31, 168, 0, 7, 25, 112, 0, 99, 0, 38, 201, 8, 0, 344, 65, 132, 33, 52, 0, 612, 52, 202, 42, 9, 0, 408, 0, 10, 411, 688, 80, 162, 0, 68, 52, 272
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OFFSET

1,8


COMMENTS

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.


LINKS

Table of n, a(n) for n=1..70.
Eric Weisstein's World of Mathematics, Unimodal Sequence


FORMULA

a(n) + A332741(n) = A318762(n).


EXAMPLE

The a(n) permutations for n = 6, 8, 9, 10, 12, 14, 15, 16:
121 132 1212 1121 1132 11121 11212 1243
231 1221 1211 1213 11211 11221 1324
2121 1231 12111 12112 1342
1312 12121 1423
1321 12211 1432
2131 21121 2143
2311 21211 2314
3121 2341
2413
2431
3142
3241
3412
3421
4132
4231


MATHEMATICA

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]], !unimodQ[#]&]], {n, 30}]


CROSSREFS

Dominated by A318762.
The complement of the nonnegated version is counted by A332294.
The nonnegated version is A332672.
The complement is counted by A332741.
A less interesting version is A333146.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Nonunimodal normal sequences are A328509.
Partitions with nonunimodal 0appended first differences are A332284.
Compositions whose negation is unimodal are A332578.
Partitions with nonunimodal negated runlengths are A332639.
Numbers whose negated prime signature is not unimodal are A332642.
Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332283, A332288, A332638, A332669, A333145.
Sequence in context: A105855 A152954 A079175 * A202815 A049336 A017888
Adjacent sequences: A332739 A332740 A332741 * A332743 A332744 A332745


KEYWORD

nonn


AUTHOR

Gus Wiseman, Mar 09 2020


STATUS

approved



